Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper.
Prepare the squash: Cut each squash in half and scrape out the seeds. Tip: Microwave each squash for 5-10 minutes to soften the skin before cutting. Make large slits before microwaving. Drizzle 2 tablespoons of olive oil over the cut sides, season with salt and pepper, and place cut-side down on the baking sheet.
Roast the squash: Place in the oven and roast for 30-40 minutes, until soft. Check doneness by piercing with a fork.
Blanch the broccoli: While the squash is roasting, boil water in a small saucepan and blanch the broccoli florets for 4 minutes. Drain well.
Cook the sausage: In a large skillet, heat the remaining olive oil over medium heat. Add the sausage and cook until browned, about 5 minutes. Drain excess fat, then add the blanched broccoli and stir to mix.
Stuff the squash: Once roasted, turn the squash halves cut-side up and let them cool slightly. Divide the sausage and broccoli mixture among the squash halves. Top with shredded cheese.
Final roast: Place the stuffed squash back in the oven and roast for another 10-15 minutes, until the cheese is melted and bubbly.